Resonant flights and transient superdiffusion in a time-periodic, two-dimensional flow

Abstract

Enhanced, passive transport is studied numerically in an oscillating vortex chain with stress-free boundary conditions. The long-range transport is found to be diffusive in the long-time limit with an effective diffusion coefficient D * that peaks dramatically in the vicinity of a few, well-defined resonant frequencies. Superdiffusive transients are also observed for frequencies near these resonant frequencies, with the duration of the transients diverging at the resonant frequencies. Standard analytical techniques based on the Melnikov approximation and on lobe dynamics fail to explain the behavior in the vicinity of these resonant peaks. An alternate explanation is provided, based on flights that have power-law scaling up to a maximum length that also diverges at the resonant frequencies. The long flights for frequencies near the resonant peaks occur because tracers in a lobe return (after an integer number of oscillation periods) to almost precisely the same location in the lobe of another vortex. These periodic orbits correspond to the formation -only at the resonant frequencies -of "tangle islands" within the chaotic region.

Thin front propagation in steady and unsteady cellular flows

“…For passive particles transport in the flow (16), it has been found that the eddy diffusivity coefficient D eff (ω) displays a complex behavior with resonant-like patterns [39,40], in which the values of D eff (ω) are orders of magnitude larger than the stationary-flow value, D eff (0). The physical mechanism responsible for the resonances is related to the interplay between the oscillation of the separatrices and the circulation inside the cell.…”

Synchronization of Oscillating Reactions in an Extended Fluid System

“…and maximum velocity v o . Transport in the oscillating vortex chain (i.e., with v d 0) has been studied extensively [19,20,25,26]; the drift term, however, is new and is added to model ( If v d > v o , transport is typically superdiffusive. In most cases, there are two chaotic regions, separated by an ordered region in the middle of the jet [light blue in Fig.…”

Lagrangian chaos and multiphase processes in vortex flows

“…The long-range transport of passive impurities in this system has been studied extensively and is reported on in previous articles. 3,4,6 For the experiments in which a drop of oil is followed, the computer stores only the coordinates of pixels brighter than a user-defined threshold, enabling us to collect data at a rate of 5-10 images per second for several hours (if the drop stays within the visible region) 16,17 .…”

“…We define a Lagrangian-generalized Capillary number that is based on finite-time Lyapunov exponents. We then present preliminary results of experimental and numerical studies for the stretching and breakup of drops of oil floating on the surface of a chain of oscillating vortices 3,4 , a flow whose mixing properties have been well-studied 5,6,7 . Statistics of the stretching of the drop are compared with those of the Lagrangian capillary number.…”